(*  Title:      HOL/Probability/Borel_Space.thy

     Author:     Johannes Hölzl, TU München

     Author:     Armin Heller, TU München

 *)

 header {*Borel spaces*}

 theory Borel_Space

   imports Sigma_Algebra "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"

 begin

 section "Generic Borel spaces"

 definition borel :: "'a::topological_space measure" where

   "borel = sigma UNIV {S. open S}"

 abbreviation "borel_measurable M \<equiv> measurable M borel"

 lemma in_borel_measurable:

    "f \<in> borel_measurable M \<longleftrightarrow>

     (\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"

   by (auto simp add: measurable_def borel_def)

 lemma in_borel_measurable_borel:

    "f \<in> borel_measurable M \<longleftrightarrow>

     (\<forall>S \<in> sets borel.

       f -` S \<inter> space M \<in> sets M)"

   by (auto simp add: measurable_def borel_def)

 lemma space_borel[simp]: "space borel = UNIV"

   unfolding borel_def by auto

 lemma borel_open[simp]:

   assumes "open A" shows "A \<in> sets borel"

 proof -

   have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .

   thus ?thesis unfolding borel_def by auto

 qed

 lemma borel_closed[simp]:

   assumes "closed A" shows "A \<in> sets borel"

 proof -

   have "space borel - (- A) \<in> sets borel"

     using assms unfolding closed_def by (blast intro: borel_open)

   thus ?thesis by simp

 qed

 lemma borel_comp[intro,simp]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"

   unfolding Compl_eq_Diff_UNIV by (intro Diff) auto

 lemma borel_measurable_vimage:

   fixes f :: "'a \<Rightarrow> 'x::t2_space"

   assumes borel: "f \<in> borel_measurable M"

   shows "f -` {x} \<inter> space M \<in> sets M"

 proof (cases "x \<in> f ` space M")

   case True then obtain y where "x = f y" by auto

   from closed_singleton[of "f y"]

   have "{f y} \<in> sets borel" by (rule borel_closed)

   with assms show ?thesis

     unfolding in_borel_measurable_borel `x = f y` by auto

 next

   case False hence "f -` {x} \<inter> space M = {}" by auto

   thus ?thesis by auto

 qed

 lemma borel_measurableI:

   fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"

   assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"

   shows "f \<in> borel_measurable M"

   unfolding borel_def

 proof (rule measurable_measure_of, simp_all)

   fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"

     using assms[of S] by simp

 qed

 lemma borel_singleton[simp, intro]:

   fixes x :: "'a::t1_space"

   shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel"

   proof (rule insert_in_sets)

     show "{x} \<in> sets borel"

       using closed_singleton[of x] by (rule borel_closed)

   qed simp

 lemma borel_measurable_const[simp, intro]:

   "(\<lambda>x. c) \<in> borel_measurable M"

   by auto

 lemma borel_measurable_indicator[simp, intro!]:

   assumes A: "A \<in> sets M"

   shows "indicator A \<in> borel_measurable M"

   unfolding indicator_def [abs_def] using A

   by (auto intro!: measurable_If_set)

 lemma borel_measurable_indicator_iff:

   "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"

     (is "?I \<in> borel_measurable M \<longleftrightarrow> _")

 proof

   assume "?I \<in> borel_measurable M"

   then have "?I -` {1} \<inter> space M \<in> sets M"

     unfolding measurable_def by auto

   also have "?I -` {1} \<inter> space M = A \<inter> space M"

     unfolding indicator_def [abs_def] by auto

   finally show "A \<inter> space M \<in> sets M" .

 next

   assume "A \<inter> space M \<in> sets M"

   moreover have "?I \<in> borel_measurable M \<longleftrightarrow>

     (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"

     by (intro measurable_cong) (auto simp: indicator_def)

   ultimately show "?I \<in> borel_measurable M" by auto

 qed

 lemma borel_measurable_subalgebra:

   assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"

   shows "f \<in> borel_measurable M"

   using assms unfolding measurable_def by auto

 section "Borel spaces on euclidean spaces"

 lemma lessThan_borel[simp, intro]:

   fixes a :: "'a\<Colon>ordered_euclidean_space"

   shows "{..< a} \<in> sets borel"

   by (blast intro: borel_open)

 lemma greaterThan_borel[simp, intro]:

   fixes a :: "'a\<Colon>ordered_euclidean_space"

   shows "{a <..} \<in> sets borel"

   by (blast intro: borel_open)

 lemma greaterThanLessThan_borel[simp, intro]:

   fixes a b :: "'a\<Colon>ordered_euclidean_space"

   shows "{a<..<b} \<in> sets borel"

   by (blast intro: borel_open)

 lemma atMost_borel[simp, intro]:

   fixes a :: "'a\<Colon>ordered_euclidean_space"

   shows "{..a} \<in> sets borel"

   by (blast intro: borel_closed)

 lemma atLeast_borel[simp, intro]:

   fixes a :: "'a\<Colon>ordered_euclidean_space"

   shows "{a..} \<in> sets borel"

   by (blast intro: borel_closed)

 lemma atLeastAtMost_borel[simp, intro]:

   fixes a b :: "'a\<Colon>ordered_euclidean_space"

   shows "{a..b} \<in> sets borel"

   by (blast intro: borel_closed)

 lemma greaterThanAtMost_borel[simp, intro]:

   fixes a b :: "'a\<Colon>ordered_euclidean_space"

   shows "{a<..b} \<in> sets borel"

   unfolding greaterThanAtMost_def by blast

 lemma atLeastLessThan_borel[simp, intro]:

   fixes a b :: "'a\<Colon>ordered_euclidean_space"

   shows "{a..<b} \<in> sets borel"

   unfolding atLeastLessThan_def by blast

 lemma hafspace_less_borel[simp, intro]:

   fixes a :: real

   shows "{x::'a::euclidean_space. a < x $$ i} \<in> sets borel"

   by (auto intro!: borel_open open_halfspace_component_gt)

 lemma hafspace_greater_borel[simp, intro]:

   fixes a :: real

   shows "{x::'a::euclidean_space. x $$ i < a} \<in> sets borel"

   by (auto intro!: borel_open open_halfspace_component_lt)

 lemma hafspace_less_eq_borel[simp, intro]:

   fixes a :: real

   shows "{x::'a::euclidean_space. a \<le> x $$ i} \<in> sets borel"

   by (auto intro!: borel_closed closed_halfspace_component_ge)

 lemma hafspace_greater_eq_borel[simp, intro]:

   fixes a :: real

   shows "{x::'a::euclidean_space. x $$ i \<le> a} \<in> sets borel"

   by (auto intro!: borel_closed closed_halfspace_component_le)

 lemma borel_measurable_less[simp, intro]:

   fixes f :: "'a \<Rightarrow> real"

   assumes f: "f \<in> borel_measurable M"

   assumes g: "g \<in> borel_measurable M"

   shows "{w \<in> space M. f w < g w} \<in> sets M"

 proof -

   have "{w \<in> space M. f w < g w} =

         (\<Union>r. (f -` {..< of_rat r} \<inter> space M) \<inter> (g -` {of_rat r <..} \<inter> space M))"

     using Rats_dense_in_real by (auto simp add: Rats_def)

   then show ?thesis using f g

     by simp (blast intro: measurable_sets)

 qed

 lemma borel_measurable_le[simp, intro]:

   fixes f :: "'a \<Rightarrow> real"

   assumes f: "f \<in> borel_measurable M"

   assumes g: "g \<in> borel_measurable M"

   shows "{w \<in> space M. f w \<le> g w} \<in> sets M"

 proof -

   have "{w \<in> space M. f w \<le> g w} = space M - {w \<in> space M. g w < f w}"

     by auto

   thus ?thesis using f g

     by simp blast

 qed

 lemma borel_measurable_eq[simp, intro]:

   fixes f :: "'a \<Rightarrow> real"

   assumes f: "f \<in> borel_measurable M"

   assumes g: "g \<in> borel_measurable M"

   shows "{w \<in> space M. f w = g w} \<in> sets M"

 proof -

   have "{w \<in> space M. f w = g w} =

         {w \<in> space M. f w \<le> g w} \<inter> {w \<in> space M. g w \<le> f w}"

     by auto

   thus ?thesis using f g by auto

 qed

 lemma borel_measurable_neq[simp, intro]:

   fixes f :: "'a \<Rightarrow> real"

   assumes f: "f \<in> borel_measurable M"

   assumes g: "g \<in> borel_measurable M"

   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"

 proof -

   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}"

     by auto

   thus ?thesis using f g by auto

 qed

 subsection "Borel space equals sigma algebras over intervals"

 lemma rational_boxes:

   fixes x :: "'a\<Colon>ordered_euclidean_space"

   assumes "0 < e"

   shows "\<exists>a b. (\<forall>i. a $$ i \<in> \<rat>) \<and> (\<forall>i. b $$ i \<in> \<rat>) \<and> x \<in> {a <..< b} \<and> {a <..< b} \<subseteq> ball x e"

 proof -

   def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"

   then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)

   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x $$ i \<and> x $$ i - y < e'" (is "\<forall>i. ?th i")

   proof

     fix i from Rats_dense_in_real[of "x $$ i - e'" "x $$ i"] e

     show "?th i" by auto

   qed

   from choice[OF this] guess a .. note a = this

   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x $$ i < y \<and> y - x $$ i < e'" (is "\<forall>i. ?th i")

   proof

     fix i from Rats_dense_in_real[of "x $$ i" "x $$ i + e'"] e

     show "?th i" by auto

   qed

   from choice[OF this] guess b .. note b = this

   { fix y :: 'a assume *: "Chi a < y" "y < Chi b"

     have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x $$ i) (y $$ i))\<twosuperior>)"

       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

     also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"

     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

       fix i assume i: "i \<in> {..<DIM('a)}"

       have "a i < y$$i \<and> y$$i < b i" using * i eucl_less[where 'a='a] by auto

       moreover have "a i < x$$i" "x$$i - a i < e'" using a by auto

       moreover have "x$$i < b i" "b i - x$$i < e'" using b by auto

       ultimately have "\<bar>x$$i - y$$i\<bar> < 2 * e'" by auto

       then have "dist (x $$ i) (y $$ i) < e/sqrt (real (DIM('a)))"

         unfolding e'_def by (auto simp: dist_real_def)

       then have "(dist (x $$ i) (y $$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"

         by (rule power_strict_mono) auto

       then show "(dist (x $$ i) (y $$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"

         by (simp add: power_divide)

     qed auto

     also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)

     finally have "dist x y < e" . }

   with a b show ?thesis

     apply (rule_tac exI[of _ "Chi a"])

     apply (rule_tac exI[of _ "Chi b"])

     using eucl_less[where 'a='a] by auto

 qed

 lemma ex_rat_list:

   fixes x :: "'a\<Colon>ordered_euclidean_space"

   assumes "\<And> i. x $$ i \<in> \<rat>"

   shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x $$ i)"

 proof -

   have "\<forall>i. \<exists>r. x $$ i = of_rat r" using assms unfolding Rats_def by blast

   from choice[OF this] guess r ..

   then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])

 qed

 lemma open_UNION:

   fixes M :: "'a\<Colon>ordered_euclidean_space set"

   assumes "open M"

   shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}

                    (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"

     (is "M = UNION ?idx ?box")

 proof safe

   fix x assume "x \<in> M"

   obtain e where e: "e > 0" "ball x e \<subseteq> M"

     using openE[OF assms `x \<in> M`] by auto

   then obtain a b where ab: "x \<in> {a <..< b}" "\<And>i. a $$ i \<in> \<rat>" "\<And>i. b $$ i \<in> \<rat>" "{a <..< b} \<subseteq> ball x e"

     using rational_boxes[OF e(1)] by blast

   then obtain p q where pq: "length p = DIM ('a)"

                             "length q = DIM ('a)"

                             "\<forall> i < DIM ('a). of_rat (p ! i) = a $$ i \<and> of_rat (q ! i) = b $$ i"

     using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast

   hence p: "Chi (of_rat \<circ> op ! p) = a"

     using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]

     unfolding o_def by auto

   from pq have q: "Chi (of_rat \<circ> op ! q) = b"

     using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]

     unfolding o_def by auto

   have "x \<in> ?box (p, q)"

     using p q ab by auto

   thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto

 qed auto

 lemma borel_sigma_sets_subset:

   "A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel"

   using sigma_sets_subset[of A borel] by simp

 lemma borel_eq_sigmaI1:

   fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"

   assumes borel_eq: "borel = sigma UNIV X"

   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range F))"

   assumes F: "\<And>i. F i \<in> sets borel"

   shows "borel = sigma UNIV (range F)"

   unfolding borel_def

 proof (intro sigma_eqI antisym)

   have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel"

     unfolding borel_def by simp

   also have "\<dots> = sigma_sets UNIV X"

     unfolding borel_eq by simp

   also have "\<dots> \<subseteq> sigma_sets UNIV (range F)"

     using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto

   finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (range F)" .

   show "sigma_sets UNIV (range F) \<subseteq> sigma_sets UNIV {S. open S}"

     unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto

 qed auto

 lemma borel_eq_sigmaI2:

   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set"

     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"

   assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"

   assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"

   assumes F: "\<And>i j. F i j \<in> sets borel"

   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"

   using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F="(\<lambda>(i, j). F i j)"]) auto

 lemma borel_eq_sigmaI3:

   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set"

   assumes borel_eq: "borel = sigma UNIV X"

   assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"

   assumes F: "\<And>i j. F i j \<in> sets borel"

   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"

   using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto

 lemma borel_eq_sigmaI4:

   fixes F :: "'i \<Rightarrow> 'a::topological_space set"

     and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set"

   assumes borel_eq: "borel = sigma UNIV (range (\<lambda>(i, j). G i j))"

   assumes X: "\<And>i j. G i j \<in> sets (sigma UNIV (range F))"

   assumes F: "\<And>i. F i \<in> sets borel"

   shows "borel = sigma UNIV (range F)"

   using assms by (intro borel_eq_sigmaI1[where X="range (\<lambda>(i, j). G i j)" and F=F]) auto

 lemma borel_eq_sigmaI5:

   fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set"

   assumes borel_eq: "borel = sigma UNIV (range G)"

   assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"

   assumes F: "\<And>i j. F i j \<in> sets borel"

   shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"

   using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto

 lemma halfspace_gt_in_halfspace:

   "{x\<Colon>'a. a < x $$ i} \<in> sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))"

   (is "?set \<in> ?SIGMA")

 proof -

   interpret sigma_algebra UNIV ?SIGMA

     by (intro sigma_algebra_sigma_sets) simp_all

   have *: "?set = (\<Union>n. UNIV - {x\<Colon>'a. x $$ i < a + 1 / real (Suc n)})"

   proof (safe, simp_all add: not_less)

     fix x assume "a < x $$ i"

     with reals_Archimedean[of "x $$ i - a"]

     obtain n where "a + 1 / real (Suc n) < x $$ i"

       by (auto simp: inverse_eq_divide field_simps)

     then show "\<exists>n. a + 1 / real (Suc n) \<le> x $$ i"

       by (blast intro: less_imp_le)

   next

     fix x n

     have "a < a + 1 / real (Suc n)" by auto

     also assume "\<dots> \<le> x"

     finally show "a < x" .

   qed

   show "?set \<in> ?SIGMA" unfolding *

     by (auto intro!: Diff)

 qed

 lemma borel_eq_halfspace_less:

   "borel = sigma UNIV (range (\<lambda>(a, i). {x::'a::ordered_euclidean_space. x $$ i < a}))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI3[OF borel_def])

   fix S :: "'a set" assume "S \<in> {S. open S}"

   then have "open S" by simp

   from open_UNION[OF this]

   obtain I where *: "S =

     (\<Union>(a, b)\<in>I.

         (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) $$ i < x $$ i}) \<inter>

         (\<Inter> i<DIM('a). {x. x $$ i < (Chi (real_of_rat \<circ> op ! b)::'a) $$ i}))"

     unfolding greaterThanLessThan_def

     unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]

     unfolding eucl_lessThan_eq_halfspaces[where 'a='a]

     by blast

   show "S \<in> ?SIGMA"

     unfolding *

     by (safe intro!: countable_UN Int countable_INT) (auto intro!: halfspace_gt_in_halfspace)

 qed auto

 lemma borel_eq_halfspace_le:

   "borel = sigma UNIV (range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x $$ i \<le> a}))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])

   fix a i

   have *: "{x::'a. x$$i < a} = (\<Union>n. {x. x$$i \<le> a - 1/real (Suc n)})"

   proof (safe, simp_all)

     fix x::'a assume *: "x$$i < a"

     with reals_Archimedean[of "a - x$$i"]

     obtain n where "x $$ i < a - 1 / (real (Suc n))"

       by (auto simp: field_simps inverse_eq_divide)

     then show "\<exists>n. x $$ i \<le> a - 1 / (real (Suc n))"

       by (blast intro: less_imp_le)

   next

     fix x::'a and n

     assume "x$$i \<le> a - 1 / real (Suc n)"

     also have "\<dots> < a" by auto

     finally show "x$$i < a" .

   qed

   show "{x. x$$i < a} \<in> ?SIGMA" unfolding *

     by (safe intro!: countable_UN) auto

 qed auto

 lemma borel_eq_halfspace_ge:

   "borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x $$ i}))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less])

   fix a i have *: "{x::'a. x$$i < a} = space ?SIGMA - {x::'a. a \<le> x$$i}" by auto

   show "{x. x$$i < a} \<in> ?SIGMA" unfolding *

       by (safe intro!: compl_sets) auto

 qed auto

 lemma borel_eq_halfspace_greater:

   "borel = sigma UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a < x $$ i}))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le])

   fix a i have *: "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto

   show "{x. x$$i \<le> a} \<in> ?SIGMA" unfolding *

     by (safe intro!: compl_sets) auto

 qed auto

 lemma borel_eq_atMost:

   "borel = sigma UNIV (range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space}))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])

   fix a i show "{x. x$$i \<le> a} \<in> ?SIGMA"

   proof cases

     assume "i < DIM('a)"

     then have *: "{x::'a. x$$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"

     proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)

       fix x

       from real_arch_simple[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"] guess k::nat ..

       then have "\<And>i. i < DIM('a) \<Longrightarrow> x$$i \<le> real k"

         by (subst (asm) Max_le_iff) auto

       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia \<le> real k"

         by (auto intro!: exI[of _ k])

     qed

     show "{x. x$$i \<le> a} \<in> ?SIGMA" unfolding *

       by (safe intro!: countable_UN) auto

   qed (auto intro: sigma_sets_top sigma_sets.Empty)

 qed auto

 lemma borel_eq_greaterThan:

   "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {a<..}))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le])

   fix a i show "{x. x$$i \<le> a} \<in> ?SIGMA"

   proof cases

     assume "i < DIM('a)"

     have "{x::'a. x$$i \<le> a} = UNIV - {x::'a. a < x$$i}" by auto

     also have *: "{x::'a. a < x$$i} = (\<Union>k::nat. {(\<chi>\<chi> n. if n = i then a else -real k) <..})" using `i <DIM('a)`

     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)

       fix x

       from reals_Archimedean2[of "Max ((\<lambda>i. -x$$i)`{..<DIM('a)})"]

       guess k::nat .. note k = this

       { fix i assume "i < DIM('a)"

         then have "-x$$i < real k"

           using k by (subst (asm) Max_less_iff) auto

         then have "- real k < x$$i" by simp }

       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x $$ ia"

         by (auto intro!: exI[of _ k])

     qed

     finally show "{x. x$$i \<le> a} \<in> ?SIGMA"

       apply (simp only:)

       apply (safe intro!: countable_UN Diff)

       apply (auto intro: sigma_sets_top)

       done

   qed (auto intro: sigma_sets_top sigma_sets.Empty)

 qed auto

 lemma borel_eq_lessThan:

   "borel = sigma UNIV (range (\<lambda>a\<Colon>'a\<Colon>ordered_euclidean_space. {..<a}))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge])

   fix a i show "{x. a \<le> x$$i} \<in> ?SIGMA"

   proof cases

     fix a i assume "i < DIM('a)"

     have "{x::'a. a \<le> x$$i} = UNIV - {x::'a. x$$i < a}" by auto

     also have *: "{x::'a. x$$i < a} = (\<Union>k::nat. {..< (\<chi>\<chi> n. if n = i then a else real k)})" using `i <DIM('a)`

     proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)

       fix x

       from reals_Archimedean2[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"]

       guess k::nat .. note k = this

       { fix i assume "i < DIM('a)"

         then have "x$$i < real k"

           using k by (subst (asm) Max_less_iff) auto

         then have "x$$i < real k" by simp }

       then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia < real k"

         by (auto intro!: exI[of _ k])

     qed

     finally show "{x. a \<le> x$$i} \<in> ?SIGMA"

       apply (simp only:)

       apply (safe intro!: countable_UN Diff)

       apply (auto intro: sigma_sets_top)

       done

   qed (auto intro: sigma_sets_top sigma_sets.Empty)

 qed auto

 lemma borel_eq_atLeastAtMost:

   "borel = sigma UNIV (range (\<lambda>(a,b). {a..b} \<Colon>'a\<Colon>ordered_euclidean_space set))"

   (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI5[OF borel_eq_atMost])

   fix a::'a

   have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"

   proof (safe, simp_all add: eucl_le[where 'a='a])

     fix x

     from real_arch_simple[of "Max ((\<lambda>i. - x$$i)`{..<DIM('a)})"]

     guess k::nat .. note k = this

     { fix i assume "i < DIM('a)"

       with k have "- x$$i \<le> real k"

         by (subst (asm) Max_le_iff) (auto simp: field_simps)

       then have "- real k \<le> x$$i" by simp }

     then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x $$ i"

       by (auto intro!: exI[of _ k])

   qed

   show "{..a} \<in> ?SIGMA" unfolding *

     by (safe intro!: countable_UN)

        (auto intro!: sigma_sets_top)

 qed auto

 lemma borel_eq_greaterThanLessThan:

   "borel = sigma UNIV (range (\<lambda> (a, b). {a <..< b} :: 'a \<Colon> ordered_euclidean_space set))"

     (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI1[OF borel_def])

   fix M :: "'a set" assume "M \<in> {S. open S}"

   then have "open M" by simp

   show "M \<in> ?SIGMA"

     apply (subst open_UNION[OF `open M`])

     apply (safe intro!: countable_UN)

     apply auto

     done

 qed auto

 lemma borel_eq_atLeastLessThan:

   "borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA")

 proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan])

   have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto

   fix x :: real

   have "{..<x} = (\<Union>i::nat. {-real i ..< x})"

     by (auto simp: move_uminus real_arch_simple)

   then show "{..< x} \<in> ?SIGMA"

     by (auto intro: sigma_sets.intros)

 qed auto

 lemma borel_measurable_halfspacesI:

   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"

   assumes F: "borel = sigma UNIV (range F)"

   and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M" 

   and S: "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"

   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"

 proof safe

   fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"

   then show "S a i \<in> sets M" unfolding assms

     by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1))

 next

   assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"

   { fix a i have "S a i \<in> sets M"

     proof cases

       assume "i < DIM('c)"

       with a show ?thesis unfolding assms(2) by simp

     next

       assume "\<not> i < DIM('c)"

       from S[OF this] show ?thesis .

     qed }

   then show "f \<in> borel_measurable M"

     by (auto intro!: measurable_measure_of simp: S_eq F)

 qed

 lemma borel_measurable_iff_halfspace_le:

   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"

   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i \<le> a} \<in> sets M)"

   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto

 lemma borel_measurable_iff_halfspace_less:

   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"

   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i < a} \<in> sets M)"

   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto

 lemma borel_measurable_iff_halfspace_ge:

   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"

   shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w $$ i} \<in> sets M)"

   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto

 lemma borel_measurable_iff_halfspace_greater:

   fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"

   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w $$ i} \<in> sets M)"

   by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto

 lemma borel_measurable_iff_le:

   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"

   using borel_measurable_iff_halfspace_le[where 'c=real] by simp

 lemma borel_measurable_iff_less:

   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"

   using borel_measurable_iff_halfspace_less[where 'c=real] by simp

 lemma borel_measurable_iff_ge:

   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"

   using borel_measurable_iff_halfspace_ge[where 'c=real] by simp

 lemma borel_measurable_iff_greater:

   "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"

   using borel_measurable_iff_halfspace_greater[where 'c=real] by simp

 lemma borel_measurable_euclidean_component:

   "(\<lambda>x::'a::euclidean_space. x $$ i) \<in> borel_measurable borel"

 proof (rule borel_measurableI)

   fix S::"real set" assume "open S"

   from open_vimage_euclidean_component[OF this]

   show "(\<lambda>x. x $$ i) -` S \<inter> space borel \<in> sets borel"

     by (auto intro: borel_open)

 qed

 lemma borel_measurable_euclidean_space:

   fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"

   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M)"

 proof safe

   fix i assume "f \<in> borel_measurable M"

   then show "(\<lambda>x. f x $$ i) \<in> borel_measurable M"

     using measurable_comp[of f _ _ "\<lambda>x. x $$ i", unfolded comp_def]

     by (auto intro: borel_measurable_euclidean_component)

 next

   assume f: "\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M"

   then show "f \<in> borel_measurable M"

     unfolding borel_measurable_iff_halfspace_le by auto

 qed

 subsection "Borel measurable operators"

 lemma affine_borel_measurable_vector:

   fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"

   assumes "f \<in> borel_measurable M"

   shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"

 proof (rule borel_measurableI)

   fix S :: "'x set" assume "open S"

   show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"

   proof cases

     assume "b \<noteq> 0"

     with `open S` have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")

       by (auto intro!: open_affinity simp: scaleR_add_right)

     hence "?S \<in> sets borel" by auto

     moreover

     from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"

       apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)

     ultimately show ?thesis using assms unfolding in_borel_measurable_borel

       by auto

   qed simp

 qed

 lemma affine_borel_measurable:

   fixes g :: "'a \<Rightarrow> real"

   assumes g: "g \<in> borel_measurable M"

   shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M"

   using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute)

 lemma borel_measurable_add[simp, intro]:

   fixes f :: "'a \<Rightarrow> real"

   assumes f: "f \<in> borel_measurable M"

   assumes g: "g \<in> borel_measurable M"

   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"

 proof -

   have 1: "\<And>a. {w\<in>space M. a \<le> f w + g w} = {w \<in> space M. a + g w * -1 \<le> f w}"

     by auto

   have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M"

     by (rule affine_borel_measurable [OF g])

   then have "\<And>a. {w \<in> space M. (\<lambda>w. a + (g w) * -1)(w) \<le> f w} \<in> sets M" using f

     by auto

   then have "\<And>a. {w \<in> space M. a \<le> f w + g w} \<in> sets M"

     by (simp add: 1)

   then show ?thesis

     by (simp add: borel_measurable_iff_ge)

 qed

 lemma borel_measurable_setsum[simp, intro]:

   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"

   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"

   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"

 proof cases

   assume "finite S"

   thus ?thesis using assms by induct auto

 qed simp

 lemma borel_measurable_square:

   fixes f :: "'a \<Rightarrow> real"

   assumes f: "f \<in> borel_measurable M"

   shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M"

 proof -

   {

     fix a

     have "{w \<in> space M. (f w)\<twosuperior> \<le> a} \<in> sets M"

     proof (cases rule: linorder_cases [of a 0])

       case less

       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = {}"

         by auto (metis less order_le_less_trans power2_less_0)

       also have "... \<in> sets M"

         by (rule empty_sets)

       finally show ?thesis .

     next

       case equal

       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =

              {w \<in> space M. f w \<le> 0} \<inter> {w \<in> space M. 0 \<le> f w}"

         by auto

       also have "... \<in> sets M"

         apply (insert f)

         apply (rule Int)

         apply (simp add: borel_measurable_iff_le)

         apply (simp add: borel_measurable_iff_ge)

         done

       finally show ?thesis .

     next

       case greater

       have "\<forall>x. (f x ^ 2 \<le> sqrt a ^ 2) = (- sqrt a  \<le> f x & f x \<le> sqrt a)"

         by (metis abs_le_interval_iff abs_of_pos greater real_sqrt_abs

                   real_sqrt_le_iff real_sqrt_power)

       hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =

              {w \<in> space M. -(sqrt a) \<le> f w} \<inter> {w \<in> space M. f w \<le> sqrt a}"

         using greater by auto

       also have "... \<in> sets M"

         apply (insert f)

         apply (rule Int)

         apply (simp add: borel_measurable_iff_ge)

         apply (simp add: borel_measurable_iff_le)

         done

       finally show ?thesis .

     qed

   }

   thus ?thesis by (auto simp add: borel_measurable_iff_le)

 qed

 lemma times_eq_sum_squares:

    fixes x::real

    shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4"

 by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric])

 lemma borel_measurable_uminus[simp, intro]:

   fixes g :: "'a \<Rightarrow> real"

   assumes g: "g \<in> borel_measurable M"

   shows "(\<lambda>x. - g x) \<in> borel_measurable M"

 proof -

   have "(\<lambda>x. - g x) = (\<lambda>x. 0 + (g x) * -1)"

     by simp

   also have "... \<in> borel_measurable M"

     by (fast intro: affine_borel_measurable g)

   finally show ?thesis .

 qed

 lemma borel_measurable_times[simp, intro]:

   fixes f :: "'a \<Rightarrow> real"

   assumes f: "f \<in> borel_measurable M"

   assumes g: "g \<in> borel_measurable M"

   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"

 proof -

   have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M"

     using assms by (fast intro: affine_borel_measurable borel_measurable_square)

   have "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) =

         (\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))"

     by (simp add: minus_divide_right)

   also have "... \<in> borel_measurable M"

     using f g by (fast intro: affine_borel_measurable borel_measurable_square f g)

   finally have 2: "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) \<in> borel_measurable M" .

   show ?thesis

     apply (simp add: times_eq_sum_squares diff_minus)

     using 1 2 by simp

 qed

 lemma borel_measurable_setprod[simp, intro]:

   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"

   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"

   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"

 proof cases

   assume "finite S"

   thus ?thesis using assms by induct auto

 qed simp

 lemma borel_measurable_diff[simp, intro]:

   fixes f :: "'a \<Rightarrow> real"

   assumes f: "f \<in> borel_measurable M"

   assumes g: "g \<in> borel_measurable M"

   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"

   unfolding diff_minus using assms by fast

 lemma borel_measurable_inverse[simp, intro]:

   fixes f :: "'a \<Rightarrow> real"

   assumes "f \<in> borel_measurable M"

   shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"

   unfolding borel_measurable_iff_ge unfolding inverse_eq_divide

 proof safe

   fix a :: real

   have *: "{w \<in> space M. a \<le> 1 / f w} =

       ({w \<in> space M. 0 < f w} \<inter> {w \<in> space M. a * f w \<le> 1}) \<union>

       ({w \<in> space M. f w < 0} \<inter> {w \<in> space M. 1 \<le> a * f w}) \<union>

       ({w \<in> space M. f w = 0} \<inter> {w \<in> space M. a \<le> 0})" by (auto simp: le_divide_eq)

   show "{w \<in> space M. a \<le> 1 / f w} \<in> sets M" using assms unfolding *

     by (auto intro!: Int Un)

 qed

 lemma borel_measurable_divide[simp, intro]:

   fixes f :: "'a \<Rightarrow> real"

   assumes "f \<in> borel_measurable M"

   and "g \<in> borel_measurable M"

   shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"

   unfolding field_divide_inverse

   by (rule borel_measurable_inverse borel_measurable_times assms)+

 lemma borel_measurable_max[intro, simp]:

   fixes f g :: "'a \<Rightarrow> real"

   assumes "f \<in> borel_measurable M"

   assumes "g \<in> borel_measurable M"

   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"

   unfolding borel_measurable_iff_le

 proof safe

   fix a

   have "{x \<in> space M. max (g x) (f x) \<le> a} =

     {x \<in> space M. g x \<le> a} \<inter> {x \<in> space M. f x \<le> a}" by auto

   thus "{x \<in> space M. max (g x) (f x) \<le> a} \<in> sets M"

     using assms unfolding borel_measurable_iff_le

     by (auto intro!: Int)

 qed

 lemma borel_measurable_min[intro, simp]:

   fixes f g :: "'a \<Rightarrow> real"

   assumes "f \<in> borel_measurable M"

   assumes "g \<in> borel_measurable M"

   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"

   unfolding borel_measurable_iff_ge

 proof safe

   fix a

   have "{x \<in> space M. a \<le> min (g x) (f x)} =

     {x \<in> space M. a \<le> g x} \<inter> {x \<in> space M. a \<le> f x}" by auto

   thus "{x \<in> space M. a \<le> min (g x) (f x)} \<in> sets M"

     using assms unfolding borel_measurable_iff_ge

     by (auto intro!: Int)

 qed

 lemma borel_measurable_abs[simp, intro]:

   assumes "f \<in> borel_measurable M"

   shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"

 proof -

   have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def)

   show ?thesis unfolding * using assms by auto

 qed

 lemma borel_measurable_nth[simp, intro]:

   "(\<lambda>x::real^'n. x $ i) \<in> borel_measurable borel"

   using borel_measurable_euclidean_component

   unfolding nth_conv_component by auto

 lemma borel_measurable_continuous_on1:

   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"

   assumes "continuous_on UNIV f"

   shows "f \<in> borel_measurable borel"

   apply(rule borel_measurableI)

   using continuous_open_preimage[OF assms] unfolding vimage_def by auto

 lemma borel_measurable_continuous_on:

   fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"

   assumes cont: "continuous_on A f" "open A"

   shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _")

 proof (rule borel_measurableI)

   fix S :: "'b set" assume "open S"

   then have "open {x\<in>A. f x \<in> S}"

     by (intro continuous_open_preimage[OF cont]) auto

   then have *: "{x\<in>A. f x \<in> S} \<in> sets borel" by auto

   have "?f -` S \<inter> space borel = 

     {x\<in>A. f x \<in> S} \<union> (if c \<in> S then space borel - A else {})"

     by (auto split: split_if_asm)

   also have "\<dots> \<in> sets borel"

     using * `open A` by (auto simp del: space_borel intro!: Un)

   finally show "?f -` S \<inter> space borel \<in> sets borel" .

 qed

 lemma convex_measurable:

   fixes a b :: real

   assumes X: "X \<in> borel_measurable M" "X ` space M \<subseteq> { a <..< b}"

   assumes q: "convex_on { a <..< b} q"

   shows "q \<circ> X \<in> borel_measurable M"

 proof -

   have "(\<lambda>x. if x \<in> {a <..< b} then q x else 0) \<in> borel_measurable borel"

   proof (rule borel_measurable_continuous_on)

     show "open {a<..<b}" by auto

     from this q show "continuous_on {a<..<b} q"

       by (rule convex_on_continuous)

   qed

   then have "(\<lambda>x. if x \<in> {a <..< b} then q x else 0) \<circ> X \<in> borel_measurable M" (is ?qX)

     using X by (intro measurable_comp) auto

   moreover have "?qX \<longleftrightarrow> q \<circ> X \<in> borel_measurable M"

     using X by (intro measurable_cong) auto

   ultimately show ?thesis by simp

 qed

 lemma borel_measurable_borel_log: assumes "1 < b" shows "log b \<in> borel_measurable borel"

 proof -

   { fix x :: real assume x: "x \<le> 0"

     { fix x::real assume "x \<le> 0" then have "\<And>u. exp u = x \<longleftrightarrow> False" by auto }

     from this[of x] x this[of 0] have "log b 0 = log b x"

       by (auto simp: ln_def log_def) }

   note log_imp = this

   have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) \<in> borel_measurable borel"

   proof (rule borel_measurable_continuous_on)

     show "continuous_on {0<..} (log b)"

       by (auto intro!: continuous_at_imp_continuous_on DERIV_log DERIV_isCont

                simp: continuous_isCont[symmetric])

     show "open ({0<..}::real set)" by auto

   qed

   also have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) = log b"

     by (simp add: fun_eq_iff not_less log_imp)

   finally show ?thesis .

 qed

 lemma borel_measurable_log[simp,intro]:

   assumes f: "f \<in> borel_measurable M" and "1 < b"

   shows "(\<lambda>x. log b (f x)) \<in> borel_measurable M"

   using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]]

   by (simp add: comp_def)

 subsection "Borel space on the extended reals"

 lemma borel_measurable_ereal_borel:

   "ereal \<in> borel_measurable borel"

 proof (rule borel_measurableI)

   fix X :: "ereal set" assume "open X"

   then have "open (ereal -` X \<inter> space borel)"

     by (simp add: open_ereal_vimage)

   then show "ereal -` X \<inter> space borel \<in> sets borel" by auto

 qed

 lemma borel_measurable_ereal[simp, intro]:

   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"

   using measurable_comp[OF f borel_measurable_ereal_borel] unfolding comp_def .

 lemma borel_measurable_real_of_ereal_borel:

   "(real :: ereal \<Rightarrow> real) \<in> borel_measurable borel"

 proof (rule borel_measurableI)

   fix B :: "real set" assume "open B"

   have *: "ereal -` real -` (B - {0}) = B - {0}" by auto

   have open_real: "open (real -` (B - {0}) :: ereal set)"

     unfolding open_ereal_def * using `open B` by auto

   show "(real -` B \<inter> space borel :: ereal set) \<in> sets borel"

   proof cases

     assume "0 \<in> B"

     then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0::ereal}"

       by (auto simp add: real_of_ereal_eq_0)

     then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel"

       using open_real by auto

   next

     assume "0 \<notin> B"

     then have *: "(real -` B :: ereal set) = real -` (B - {0})"

       by (auto simp add: real_of_ereal_eq_0)

     then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel"

       using open_real by auto

   qed

 qed

 lemma borel_measurable_real_of_ereal[simp, intro]:

   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. real (f x :: ereal)) \<in> borel_measurable M"

   using measurable_comp[OF f borel_measurable_real_of_ereal_borel] unfolding comp_def .

 lemma borel_measurable_ereal_iff:

   shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"

 proof

   assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"

   from borel_measurable_real_of_ereal[OF this]

   show "f \<in> borel_measurable M" by auto

 qed auto

 lemma borel_measurable_ereal_iff_real:

   fixes f :: "'a \<Rightarrow> ereal"

   shows "f \<in> borel_measurable M \<longleftrightarrow>

     ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"

 proof safe

   assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"

   have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto

   with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all

   let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real (f x))"

   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto

   also have "?f = f" by (auto simp: fun_eq_iff ereal_real)

   finally show "f \<in> borel_measurable M" .

 qed (auto intro: measurable_sets borel_measurable_real_of_ereal)

 lemma less_eq_ge_measurable:

   fixes f :: "'a \<Rightarrow> 'c::linorder"

   shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M"

 proof

   assume "f -` {a <..} \<inter> space M \<in> sets M"

   moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto

   ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto

 next

   assume "f -` {..a} \<inter> space M \<in> sets M"

   moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto

   ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto

 qed

 lemma greater_eq_le_measurable:

   fixes f :: "'a \<Rightarrow> 'c::linorder"

   shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M"

 proof

   assume "f -` {a ..} \<inter> space M \<in> sets M"

   moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto

   ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto

 next

   assume "f -` {..< a} \<inter> space M \<in> sets M"

   moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto

   ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto

 qed

 lemma borel_measurable_uminus_borel_ereal:

   "(uminus :: ereal \<Rightarrow> ereal) \<in> borel_measurable borel"

 proof (rule borel_measurableI)

   fix X :: "ereal set" assume "open X"

   have "uminus -` X = uminus ` X" by (force simp: image_iff)

   then have "open (uminus -` X)" using `open X` ereal_open_uminus by auto

   then show "uminus -` X \<inter> space borel \<in> sets borel" by auto

 qed

 lemma borel_measurable_uminus_ereal[intro]:

   assumes "f \<in> borel_measurable M"

   shows "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"

   using measurable_comp[OF assms borel_measurable_uminus_borel_ereal] by (simp add: comp_def)

 lemma borel_measurable_uminus_eq_ereal[simp]:

   "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")

 proof

   assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp

 qed auto

 lemma borel_measurable_eq_atMost_ereal:

   fixes f :: "'a \<Rightarrow> ereal"

   shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"

 proof (intro iffI allI)

   assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M"

   show "f \<in> borel_measurable M"

     unfolding borel_measurable_ereal_iff_real borel_measurable_iff_le

   proof (intro conjI allI)

     fix a :: real

     { fix x :: ereal assume *: "\<forall>i::nat. real i < x"

       have "x = \<infinity>"

       proof (rule ereal_top)

         fix B from reals_Archimedean2[of B] guess n ..

         then have "ereal B < real n" by auto

         with * show "B \<le> x" by (metis less_trans less_imp_le)

       qed }

     then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)"

       by (auto simp: not_le)

     then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos by (auto simp del: UN_simps intro!: Diff)

     moreover

     have "{-\<infinity>::ereal} = {..-\<infinity>}" by auto

     then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto

     moreover have "{x\<in>space M. f x \<le> ereal a} \<in> sets M"

       using pos[of "ereal a"] by (simp add: vimage_def Int_def conj_commute)

     moreover have "{w \<in> space M. real (f w) \<le> a} =

       (if a < 0 then {w \<in> space M. f w \<le> ereal a} - f -` {-\<infinity>} \<inter> space M

       else {w \<in> space M. f w \<le> ereal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r")

       proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed

     ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto

   qed

 qed (simp add: measurable_sets)

 lemma borel_measurable_eq_atLeast_ereal:

   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"

 proof

   assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M"

   moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}"

     by (auto simp: ereal_uminus_le_reorder)

   ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M"

     unfolding borel_measurable_eq_atMost_ereal by auto

   then show "f \<in> borel_measurable M" by simp

 qed (simp add: measurable_sets)

 lemma borel_measurable_ereal_iff_less:

   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"

   unfolding borel_measurable_eq_atLeast_ereal greater_eq_le_measurable ..

 lemma borel_measurable_ereal_iff_ge:

   "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"

   unfolding borel_measurable_eq_atMost_ereal less_eq_ge_measurable ..

 lemma borel_measurable_ereal_eq_const:

   fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"

   shows "{x\<in>space M. f x = c} \<in> sets M"

 proof -

   have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto

   then show ?thesis using assms by (auto intro!: measurable_sets)

 qed

 lemma borel_measurable_ereal_neq_const:

   fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"

   shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"

 proof -

   have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto

   then show ?thesis using assms by (auto intro!: measurable_sets)

 qed

 lemma borel_measurable_ereal_le[intro,simp]:

   fixes f g :: "'a \<Rightarrow> ereal"

   assumes f: "f \<in> borel_measurable M"

   assumes g: "g \<in> borel_measurable M"

   shows "{x \<in> space M. f x \<le> g x} \<in> sets M"

 proof -

   have "{x \<in> space M. f x \<le> g x} =

     {x \<in> space M. real (f x) \<le> real (g x)} - (f -` {\<infinity>, -\<infinity>} \<inter> space M \<union> g -` {\<infinity>, -\<infinity>} \<inter> space M) \<union>

     f -` {-\<infinity>} \<inter> space M \<union> g -` {\<infinity>} \<inter> space M" (is "?l = ?r")

   proof (intro set_eqI)

     fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases rule: ereal2_cases[of "f x" "g x"]) auto

   qed

   with f g show ?thesis by (auto intro!: Un simp: measurable_sets)

 qed

 lemma borel_measurable_ereal_less[intro,simp]:

   fixes f :: "'a \<Rightarrow> ereal"

   assumes f: "f \<in> borel_measurable M"

   assumes g: "g \<in> borel_measurable M"

   shows "{x \<in> space M. f x < g x} \<in> sets M"

 proof -

   have "{x \<in> space M. f x < g x} = space M - {x \<in> space M. g x \<le> f x}" by auto

   then show ?thesis using g f by auto

 qed

 lemma borel_measurable_ereal_eq[intro,simp]:

   fixes f :: "'a \<Rightarrow> ereal"

   assumes f: "f \<in> borel_measurable M"

   assumes g: "g \<in> borel_measurable M"

   shows "{w \<in> space M. f w = g w} \<in> sets M"

 proof -

   have "{x \<in> space M. f x = g x} = {x \<in> space M. g x \<le> f x} \<inter> {x \<in> space M. f x \<le> g x}" by auto

   then show ?thesis using g f by auto

 qed

 lemma borel_measurable_ereal_neq[intro,simp]:

   fixes f :: "'a \<Rightarrow> ereal"

   assumes f: "f \<in> borel_measurable M"

   assumes g: "g \<in> borel_measurable M"

   shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"

 proof -

   have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}" by auto

   thus ?thesis using f g by auto

 qed

 lemma split_sets:

   "{x\<in>space M. P x \<or> Q x} = {x\<in>space M. P x} \<union> {x\<in>space M. Q x}"

   "{x\<in>space M. P x \<and> Q x} = {x\<in>space M. P x} \<inter> {x\<in>space M. Q x}"

   by auto

 lemma borel_measurable_ereal_add[intro, simp]:

   fixes f :: "'a \<Rightarrow> ereal"

   assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"

   shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"

 proof -

   { fix x assume "x \<in> space M" then have "f x + g x =

       (if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>

         else if f x = -\<infinity> \<or> g x = -\<infinity> then -\<infinity>

         else ereal (real (f x) + real (g x)))"

       by (cases rule: ereal2_cases[of "f x" "g x"]) auto }

   with assms show ?thesis

     by (auto cong: measurable_cong simp: split_sets

              intro!: Un measurable_If measurable_sets)

 qed

 lemma borel_measurable_ereal_setsum[simp, intro]:

   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"

   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"

   shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"

 proof cases

   assume "finite S"

   thus ?thesis using assms

     by induct auto

 qed (simp add: borel_measurable_const)

 lemma borel_measurable_ereal_abs[intro, simp]:

   fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"

   shows "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"

 proof -

   { fix x have "\<bar>f x\<bar> = (if 0 \<le> f x then f x else - f x)" by auto }

   then show ?thesis using assms by (auto intro!: measurable_If)

 qed

 lemma borel_measurable_ereal_times[intro, simp]:

   fixes f :: "'a \<Rightarrow> ereal" assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"

   shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"

 proof -

   { fix f g :: "'a \<Rightarrow> ereal"

     assume b: "f \<in> borel_measurable M" "g \<in> borel_measurable M"

       and pos: "\<And>x. 0 \<le> f x" "\<And>x. 0 \<le> g x"

     { fix x have *: "f x * g x = (if f x = 0 \<or> g x = 0 then 0

         else if f x = \<infinity> \<or> g x = \<infinity> then \<infinity>

         else ereal (real (f x) * real (g x)))"

       apply (cases rule: ereal2_cases[of "f x" "g x"])

       using pos[of x] by auto }

     with b have "(\<lambda>x. f x * g x) \<in> borel_measurable M"

       by (auto cong: measurable_cong simp: split_sets

                intro!: Un measurable_If measurable_sets) }

   note pos_times = this

   have *: "(\<lambda>x. f x * g x) =

     (\<lambda>x. if 0 \<le> f x \<and> 0 \<le> g x \<or> f x < 0 \<and> g x < 0 then \<bar>f x\<bar> * \<bar>g x\<bar> else - (\<bar>f x\<bar> * \<bar>g x\<bar>))"

     by (auto simp: fun_eq_iff)

   show ?thesis using assms unfolding *

     by (intro measurable_If pos_times borel_measurable_uminus_ereal)

        (auto simp: split_sets intro!: Int)

 qed

 lemma borel_measurable_ereal_setprod[simp, intro]:

   fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal"

   assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"

   shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"

 proof cases

   assume "finite S"

   thus ?thesis using assms by induct auto

 qed simp

 lemma borel_measurable_ereal_min[simp, intro]:

   fixes f g :: "'a \<Rightarrow> ereal"

   assumes "f \<in> borel_measurable M"

   assumes "g \<in> borel_measurable M"

   shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"

   using assms unfolding min_def by (auto intro!: measurable_If)

 lemma borel_measurable_ereal_max[simp, intro]:

   fixes f g :: "'a \<Rightarrow> ereal"

   assumes "f \<in> borel_measurable M"

   and "g \<in> borel_measurable M"

   shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"

   using assms unfolding max_def by (auto intro!: measurable_If)

 lemma borel_measurable_SUP[simp, intro]:

   fixes f :: "'d\<Colon>countable \<Rightarrow> 'a \<Rightarrow> ereal"

   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"

   shows "(\<lambda>x. SUP i : A. f i x) \<in> borel_measurable M" (is "?sup \<in> borel_measurable M")

   unfolding borel_measurable_ereal_iff_ge

 proof

   fix a

   have "?sup -` {a<..} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. a < f i x})"

     by (auto simp: less_SUP_iff)

   then show "?sup -` {a<..} \<inter> space M \<in> sets M"

     using assms by auto

 qed

 lemma borel_measurable_INF[simp, intro]:

   fixes f :: "'d :: countable \<Rightarrow> 'a \<Rightarrow> ereal"

   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<in> borel_measurable M"

   shows "(\<lambda>x. INF i : A. f i x) \<in> borel_measurable M" (is "?inf \<in> borel_measurable M")

   unfolding borel_measurable_ereal_iff_less

 proof

   fix a

   have "?inf -` {..<a} \<inter> space M = (\<Union>i\<in>A. {x\<in>space M. f i x < a})"

     by (auto simp: INF_less_iff)

   then show "?inf -` {..<a} \<inter> space M \<in> sets M"

     using assms by auto

 qed

 lemma borel_measurable_liminf[simp, intro]:

   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"

   assumes "\<And>i. f i \<in> borel_measurable M"

   shows "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M"

   unfolding liminf_SUPR_INFI using assms by auto

 lemma borel_measurable_limsup[simp, intro]:

   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"

   assumes "\<And>i. f i \<in> borel_measurable M"

   shows "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M"

   unfolding limsup_INFI_SUPR using assms by auto

 lemma borel_measurable_ereal_diff[simp, intro]:

   fixes f g :: "'a \<Rightarrow> ereal"

   assumes "f \<in> borel_measurable M"

   assumes "g \<in> borel_measurable M"

   shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"

   unfolding minus_ereal_def using assms by auto

 lemma borel_measurable_ereal_inverse[simp, intro]:

   assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M"

 proof -

   { fix x have "inverse (f x) = (if f x = 0 then \<infinity> else ereal (inverse (real (f x))))"

       by (cases "f x") auto }

   with f show ?thesis

     by (auto intro!: measurable_If)

 qed

 lemma borel_measurable_ereal_divide[simp, intro]:

   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. f x / g x :: ereal) \<in> borel_measurable M"

   unfolding divide_ereal_def by auto

 lemma borel_measurable_psuminf[simp, intro]:

   fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"

   assumes "\<And>i. f i \<in> borel_measurable M" and pos: "\<And>i x. x \<in> space M \<Longrightarrow> 0 \<le> f i x"

   shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M"

   apply (subst measurable_cong)

   apply (subst suminf_ereal_eq_SUPR)

   apply (rule pos)

   using assms by auto

 section "LIMSEQ is borel measurable"

 lemma borel_measurable_LIMSEQ:

   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"

   assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"

   and u: "\<And>i. u i \<in> borel_measurable M"

   shows "u' \<in> borel_measurable M"

 proof -

   have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)"

     using u' by (simp add: lim_imp_Liminf)

   moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M"

     by auto

   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)

 qed

 end